<s>
In	O
mathematics	O
,	O
the	O
Menger	B-Algorithm
sponge	I-Algorithm
(	O
also	O
known	O
as	O
the	O
Menger	B-Algorithm
cube	I-Algorithm
,	O
Menger	O
universal	O
curve	O
,	O
Sierpinski	B-Algorithm
cube	I-Algorithm
,	O
or	O
Sierpinski	B-Algorithm
sponge	I-Algorithm
)	O
is	O
a	O
fractal	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
The	O
construction	O
of	O
a	O
Menger	B-Algorithm
sponge	I-Algorithm
can	O
be	O
described	O
as	O
follows	O
:	O
</s>
<s>
This	O
is	O
a	O
level-1	O
Menger	B-Algorithm
sponge	I-Algorithm
(	O
resembling	O
a	O
void	O
cube	O
)	O
.	O
</s>
<s>
The	O
Menger	B-Algorithm
sponge	I-Algorithm
itself	O
is	O
the	O
limit	O
of	O
this	O
process	O
after	O
an	O
infinite	O
number	O
of	O
iterations	O
.	O
</s>
<s>
The	O
th	O
stage	O
of	O
the	O
Menger	B-Algorithm
sponge	I-Algorithm
,	O
,	O
is	O
made	O
up	O
of	O
smaller	O
cubes	O
,	O
each	O
with	O
a	O
side	O
length	O
of	O
(	O
1/3	O
)	O
n	O
.	O
The	O
total	O
volume	O
of	O
is	O
thus	O
.	O
</s>
<s>
The	O
Lebesgue	O
covering	O
dimension	O
of	O
the	O
Menger	B-Algorithm
sponge	I-Algorithm
is	O
one	O
,	O
the	O
same	O
as	O
any	O
curve	O
.	O
</s>
<s>
Menger	O
showed	O
,	O
in	O
the	O
1926	O
construction	O
,	O
that	O
the	O
sponge	O
is	O
a	O
universal	O
curve	O
,	O
in	O
that	O
every	O
curve	O
is	O
homeomorphic	O
to	O
a	O
subset	O
of	O
the	O
Menger	B-Algorithm
sponge	I-Algorithm
,	O
where	O
a	O
curve	O
means	O
any	O
compact	O
metric	O
space	O
of	O
Lebesgue	O
covering	O
dimension	O
one	O
;	O
this	O
includes	O
trees	O
and	O
graphs	O
with	O
an	O
arbitrary	O
countable	O
number	O
of	O
edges	O
,	O
vertices	O
and	O
closed	O
loops	O
,	O
connected	O
in	O
arbitrary	O
ways	O
.	O
</s>
<s>
The	O
Menger	B-Algorithm
sponge	I-Algorithm
constructed	O
in	O
three	O
dimensions	O
extends	O
this	O
idea	O
to	O
graphs	O
that	O
are	O
not	O
planar	O
and	O
might	O
be	O
embedded	O
in	O
any	O
number	O
of	O
dimensions	O
.	O
</s>
<s>
The	O
Menger	B-Algorithm
sponge	I-Algorithm
is	O
a	O
closed	O
set	O
;	O
since	O
it	O
is	O
also	O
bounded	O
,	O
the	O
Heine	O
–	O
Borel	O
theorem	O
implies	O
that	O
it	O
is	O
compact	O
.	O
</s>
<s>
Experiments	O
also	O
showed	O
that	O
cubes	O
with	O
a	O
Menger	B-Algorithm
sponge	I-Algorithm
structure	O
could	O
dissipate	O
shocks	O
five	O
times	O
better	O
for	O
the	O
same	O
material	O
than	O
cubes	O
without	O
any	O
pores	O
.	O
</s>
<s>
Formally	O
,	O
a	O
Menger	B-Algorithm
sponge	I-Algorithm
can	O
be	O
defined	O
as	O
follows	O
:	O
</s>
<s>
MegaMenger	O
was	O
a	O
project	O
aiming	O
to	O
build	O
the	O
largest	O
fractal	B-Application
model	O
,	O
pioneered	O
by	O
Matt	O
Parker	O
of	O
Queen	O
Mary	O
University	O
of	O
London	O
and	O
Laura	O
Taalman	O
of	O
James	O
Madison	O
University	O
.	O
</s>
<s>
In	O
2014	O
,	O
twenty	O
level-three	O
Menger	B-Algorithm
sponges	I-Algorithm
were	O
constructed	O
,	O
which	O
combined	O
would	O
form	O
a	O
distributed	O
level-four	O
Menger	B-Algorithm
sponge	I-Algorithm
.	O
</s>
<s>
A	O
Jerusalem	O
cube	O
is	O
a	O
fractal	B-Application
object	O
described	O
by	O
Eric	O
Baird	O
in	O
2011	O
.	O
</s>
<s>
The	O
construction	O
is	O
similar	O
to	O
the	O
Menger	B-Algorithm
sponge	I-Algorithm
but	O
with	O
two	O
different-sized	O
cubes	O
.	O
</s>
<s>
Since	O
the	O
edge	O
length	O
of	O
a	O
cube	O
of	O
rank	O
N	O
is	O
equal	O
to	O
that	O
of	O
2	O
cubes	O
of	O
rank	O
N+1	O
and	O
a	O
cube	O
of	O
rank	O
N+2	O
,	O
it	O
follows	O
that	O
the	O
scaling	O
factor	O
must	O
satisfy	O
,	O
therefore	O
which	O
means	O
the	O
fractal	B-Application
cannot	O
be	O
constructed	O
on	O
a	O
rational	O
grid	O
.	O
</s>
<s>
As	O
with	O
the	O
Menger	B-Algorithm
sponge	I-Algorithm
,	O
the	O
faces	O
of	O
a	O
Jerusalem	O
cube	O
are	O
fractals	B-Application
with	O
the	O
same	O
scaling	O
factor	O
.	O
</s>
<s>
A	O
Mosely	O
snowflake	O
is	O
a	O
cube-based	O
fractal	B-Application
with	O
corners	O
recursively	O
removed	O
.	O
</s>
<s>
A	O
tetrix	O
is	O
a	O
tetrahedron-based	O
fractal	B-Application
made	O
from	O
four	O
smaller	O
copies	O
,	O
arranged	O
in	O
a	O
tetrahedron	O
.	O
</s>
<s>
A	O
Sierpinski	O
–	O
Menger	O
snowflake	O
is	O
a	O
cube-based	O
fractal	B-Application
in	O
which	O
eight	O
corner	O
cubes	O
and	O
one	O
central	O
cube	O
are	O
kept	O
each	O
time	O
at	O
the	O
lower	O
and	O
lower	O
recursion	O
steps	O
.	O
</s>
<s>
This	O
peculiar	O
three-dimensional	O
fractal	B-Application
has	O
the	O
Hausdorff	O
dimension	O
of	O
the	O
natively	O
two-dimensional	O
object	O
like	O
the	O
plane	O
i.e.	O
</s>
